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The geometry of anabelioids. (English) Zbl 1113.14021

The author makes a rather detailed exposition of his theory of anabelioids. The article is motivated by the question (following the introduction): “To what extent can the fundamental group of a Galois category be constructed in a canonical fashion which is independent of a choice of a basepoint?” The point of view adopted in the article is motivated by A. Grothendieck’s anabelian philosophy [in: Geometric Galois Actions. 1. Around Grothendieck’s Esquisse d’un Programme, London Math. Soc. Lect. Note Ser. 242, 49–58 (1997; Zbl 0901.14002)].
In the first two sections of the paper the author develops the general theory of anabelioids. An anabelioid is a finite product of categories \(X=\prod_iX_i\), where each \(X_i\) (a “connected component”) is a category isomorphic to some category \(B(G)\) of finite sets with a continuous \(G\)-action for some fixed profinite group \(G\). One of the central notions of the paper is the “faithful quasi-core” of an anabelioid, a weakening of the “core” defined in a previous paper of the author [J. Pure Appl. Algebra 131, 227–244 (1998; Zbl 0965.14013)]. The condition for a quasi-core is essentially that a certain forgetful functor is an equivalence.
The main result of this part of the paper is: when an anabelioid possesses a faithful quasi-core, then its fundamental group may be constructed in a canonical way as a profinite group. The last part of the paper is devoted to the case of hyperbolic curves: the theory of “cores” [S. Mochizuki, loc. cit. and Doc. Math., J. DMV Extra Vol., 609–640 (2003; Zbl 1092.14507)] in the context of hyperbolic curves is translated into the language of anabelioids. The author proves that if a non-proper hyperbolic curve over a \(p\)-adic or a number field is a geometric core, then its associated anabelioid admits a faithful quasi-core.

MSC:

14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
14H30 Coverings of curves, fundamental group
14F35 Homotopy theory and fundamental groups in algebraic geometry
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